Nonlinear Relativistic and Quantum Equations with a Common Type of Solution

Abstract

Generalizations of the three main equations of quantum physics, namely, the Schrödinger, Klein-Gordon, and Dirac equations, are proposed. Nonlinear terms, characterized by exponents depending on an index $q$, are considered in such a way that the standard linear equations are recovered in the limit $q\to 1$. Interestingly, these equations present a common, soliton-like, travelling solution, which is written in terms of the $q$-exponential function that naturally emerges within nonextensive statistical mechanics. In all cases, the well-known Einstein energy-momentum relation is preserved for arbitrary values of $q$.

References in corpus (5)

• Superdiffusion and non-Gaussian statistics in a driven-dissipative 2D dusty plasma
• Absence of Wavepacket Diffusion in Disordered Nonlinear Systems
• Power-Law Distributions for a Trapped Ion Interacting with a Classical Buffer Gas
• Consequences of the H-Theorem from Nonlinear Fokker-Planck Equations
• A General Nonlinear Fokker-Planck Equation and its Associated Entropy

Cited by in corpus (52)

• Black hole thermodynamical entropy
• Displacement operator for quantum systems with position-dependent mass
• Morse potential derived from first principles
• A position-dependent mass harmonic oscillator and deformed space
• Modulation of ion-acoustic waves in a nonextensive plasma with two-temperature electrons
• Topological Invariant and Anomalous Edge Modes of Strongly Nonlinear Systems
• Generalized space and linear momentum operators in quantum mechanics
• Reply to Comment on "Towards a large deviation theory for strongly correlated systems"
• Nonlinear Schroedinger Equation in the Presence of Uniform Acceleration
• Variational Approach and Deformed Derivatives
• Dual Conformable Derivative: Definition, Simple Properties and Perspectives for Applications
• A Family of Exact, Analytic Time Dependent Wave Packet Solutions to a Nonlinear Schroedinger Equation
• Nonlinear Landau damping and modulation of electrostatic waves in a nonextensive electron-positron-pair plasma
• Possible Divergences in Tsallis' Thermostatistics
• New features of quantum discord uncovered by q-entropies
• Consistent nonadditive approach and nuclear equation of state
• q-Moments remove the degeneracy associated with the inversion of the q-Fourier transform
• Deformed Fokker-Planck equation: inhomogeneous medium with a position-dependent mass
• Extending the D'Alembert Solution to Space-Time Modified Riemann-Liouville Fractional Wave Equations
• Generalization of the possible algebraic basis of $q$-triplets
• Perturbative Thermodynamic Geometry of Nonextensive Ideal Classical, Bose and Fermi Gases
• From the hypergeometric differential equation to a non linear Schrödinger one
• Conceptual Inadequacy of the Shore and Johnson Axioms for Wide Classes of Complex Systems
• Nonextensive Nuclear Liquid-Gas Phase Transition
• Chiral phase transition in linear sigma model with non-extensive statistical mechanics
• Boltzmann-Gibbs entropy is sufficient but not necessary for the likelihood factorization required by Einstein
• Inter-occurrence times and universal laws in finance, earthquakes and genomes
• Long-range interactions, doubling measures and Tsallis entropy
• Entanglement rates for Renyi, Tsallis and other entropies
• Generalized Euler, Smoluchowski and Schrödinger equations admitting self-similar solutions with a Tsallis invariant profile
• Solutions for a q-generalized Schrödinger equation of entangled interacting particles
• Modified entropies as the origin of generalized uncertainty principles
• Sensitivity, entropy, and escape rates at the onset of chaos
• Dissipative Effects in Nonlinear Klein-Gordon Dynamics
• Long memory constitutes a unified mesoscopic mechanism consistent with nonextensive statistical mechanics
• Generalized Nonlinear Proca Equation and its Free-Particle Solutions
• Structural scale $q-$derivative and the LLG-Equation in a scenario with fractionality
• q-Gamow States as continuous linear functionals on analytical test functions
• Hypergeometric Connotations of Quantum Equations
• Tsallis' Quantum q-Fields
• q-Gamow States for intermediate energies
• Classical and \textcolor{blue}{Quantum} Field-Theoretical approach to the non-linear q-Klein-Gordon Equation
• Along the lines of nonadditive entropies: $q$-prime numbers and $q$-zeta functions
• Generalized and new solutions of the NRT nonlinear Schrödinger equation
• Harmonic oscillators from displacement operators and thermodynamics
• Nonextensive statistical field theory
• Morse potential in relativistic contexts from generalized momentum operator, Pekeris approximation revisited and mapping
• Lorentzian Solitary Wave in a Generalised Nonlinear Schrödinger Equation
• Critical exponents and amplitude ratios of scalar nonextensive $q$-field theories
• An introduction to nonadditive entropies and a thermostatistical approach of inanimate and living matter
• Two types of $q$-Gaussian distributions used to study the diffusion in a finite region
• A Nonlinear $q$-Deformed Schrödinger Equation